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Mathematics Stack Exchange

Questions tagged [convergence-divergence]

Convergence and divergence of sequences and series and different modes of convergence and divergence. Also for convergence of improper integrals.

This tag is for questions about Convergent Sequences and Series and their existence, and by extension also for divergent ones. Also for convergence/divergence of improper integrals.

Formally, a sequence $S_n$ converges to the limit $S.$

$$\lim_{n\to \infty}S_n=S $$ if, for any $\epsilon>0$, there exists an $N>0$ such that $|S_n-S|<\epsilon$ for $n>N$.

A sequence diverges if it does not converge.

For more specialized questions about divergent series, such as summation methods, consider the tag .

21990 questions
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Finding values that make the series converge

For which values of $\theta \in [0,2\pi)$ does the sum converge? And then for these values of $\theta $, find the sum of the series. The given series for this question is $\sum_{n=0}^{\infty} (sin\theta)^n$ So this particular series is a geometric…
bjp409
  • 109
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Convergence test for $\sum\limits_{k=0}^{\infty}\frac{x^k}{k!}$

I need to explain the convergence of $\sum\limits_{k=0}^{\infty}\frac{x^k}{k!}$ So my working out I used the ratio test and I got up to $\sum\limits_{k=0}^{\infty}\frac{a}{k}+1$ Since $\sum\limits_{k=0}^{\infty}\frac{1}{k}+1=0$ Then…
Leo Lang
  • 35
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Does the sum from $1$ to infinity of $1/n^{1.01 }$ converge?

I looked this up on Wolfram Alpha and it said it converged to $100.578$. Is this correct? Much more importantly, how would I solve the question please? I am at pre-uni level (this is an old interview question) so I'm afraid even basic techniques…
John Smith
  • 628
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convergence of sequence of function in measure space

Let $(X,\mathcal{M},\mu)$ be a finite measure space. Prove that $(f_n)_n$ converges to $f$ in measure if and only if $\mathrm{d}(f_n,f)\rightarrow 0$, where $\mathrm{d}$ is the metric defined as follows: given $f,g\,\colon X\to\mathbb{C}$, then…
indrasis
  • 25
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Counterexample convergence of a sequence of functions in the $L^1$ norm vs pointwise

I have to show that $f(x) \neq \lim_{n\to\infty} f_n(x_n) $, where $x_n$ is a converging sequence in $[0,1]$, and $f_n \rightarrow f$ in the $L^1$ norm, such that $ \int_0^1 | f_n (x) - f(x) | dt < \epsilon $. So, as far as I got, I need to find a…
Mino
  • 129
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Stronger form on mean convergence implies almost everywhere convergence?

In a measure space $(X,\cal{A},\mu)$, suppose we are given functions $f$ and $f_1,f_2,\ldots$ all in $\cal{L}(X,\cal{A},\mu,\mathbb{R})$ such that a condition stronger than mean convergence follows, i.e., we assume…
mosagepa
  • 33
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Test the convergence of $\sum_{n=0}^{\infty} \frac{n^{k+1}}{n^k + k}$

Problem: Test the convergence of $\sum_{n=0}^{\infty} \frac{n^{k+1}}{n^k + k}$, where $k$ is a positive constant. I'm stumped. I've tried to apply several different convergence tests, but still can't figure this one out.
Damir
  • 123
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3 answers

Check convergence of $\sum^{\infty}_{n=1} \frac{2^{n} +n ^{2}}{3^{n} +n ^{3}}$

Zoomed version: $$\sum^{\infty}_{n=1} \frac{2^{n} +n ^{2}}{3^{n} +n ^{3}}$$ So, I've seen similar example at Convergence or divergence of $\sum \frac{3^n + n^2}{2^n + n^3}$ And I liked that answer :…
Mex
  • 91
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Sequence of nowhere differentiable functions that converges uniformly, where f is everywhere differentiable?

I was wondering if there exists a sequence $(f_n)$ of nowhere differentiable functions $(f_n) \rightarrow f$ uniformly, BUT $f$ everywhere differentiable. I have a hunch this violates a theorem of differentiability, but I can't put my tongue on…
Wow McWow
  • 31
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1 answer

Intervals of Limits of Cesàro Means

Consider an infinite sequence $(a_k)$. I am interested in the limit of Cesàro means of that sequence—i.e., $$\lim_{n\to\infty}\frac1n\sum_{k=1}^na_k$$ and of sequences that are rearrangements of the terms of $(a_k)$. For example, the consider the…
bakeryjake
  • 69
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1 answer

convergence of continued nested function

Does $$\sin (x + \sin (x + \sin (x + \sin (x + ... ) ) ))) $$ converge to any limit? If so, to what? Recursive plotting appears at times to come to same/similar profiles.
Narasimham
  • 40,495
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Are all three conditions for the Leibniz criterion needed?

I'm trying to convince myself that all three conditions \begin{equation} \tag{1} (a_n)_{n\in\mathbb{N}} \text{ is null sequence} \end{equation} \begin{equation} \tag{2} (|a_n|)_{n\in\mathbb{N}} \text{ is monotonically…
Clayton Louden
  • 155
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3 answers

Convergence of series $\sum_{n=2}^\infty\left (1-\frac 1n\right)^n$

I would like to know whether the following series $$\sum_{n=2}^\infty \left(1-\frac 1n\right)^n$$ converges. The root test and ratio test are inconclusive. And I can't apply the Weierstrass M-test...
IDontKnowMath
  • 607
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1 answer

Convergence in mean exercise

The exercise states: Show that $X_n{\buildrel 1 \over \to} X \implies E(X_n) \to E(X)$. Does the converse implication hold? $\underline{SOLUTION}$ For the first part i did the following: $|E(X_n)-E(X)| \leq\{Jensen's\hspace{1mm} inequality\} \leq…
teo theo
  • 83
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1 answer

Forming a directed system from a family of closed subsets with the finite intersection property

In the beginning of chapter 4 of Dr. Pete Clark's convergence notes: http://alpha.math.uga.edu/~pete/convergence.pdf Theorem 4.1 (page 13) asserts the equivalence of 5 conditions. After making a nice observation involving De Morgan's law and…
roo
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